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Finiteness and the Emergence of Duality
by Matilda Delgado, Damian van de Heisteeg, Sanjay Raman, Ethan Torres, Cumrun Vafa, Kai Xu
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Submission summary
Authors (as registered SciPost users): | Matilda Delgado |
Submission information | |
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Preprint Link: | scipost_202502_00029v1 (pdf) |
Date submitted: | Feb. 17, 2025, 9:34 a.m. |
Submitted by: | Delgado, Matilda |
Submitted to: | SciPost Physics |
Ontological classification | |
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Academic field: | Physics |
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Approach: | Theoretical |
Abstract
We argue that the finiteness of quantum gravity amplitudes in fully compactified theories (at least in supersymmetric cases) leads to a bottom-up prediction for the existence of non-trivial dualities. In particular, finiteness requires the moduli space of massless fields to be compactifiable, meaning that its volume must be finite or at least grow no faster than that of Euclidean space. Moreover, we relate the compactifiability of moduli spaces to the condition that the lattice of charged objects transform in a semisimple representation under the action of the duality group. These ideas are supported by a wide variety of string theory examples.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
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Nonetheless, I would suggest the authors to address the following minor comments:
Requested changes
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In section 2.1.1 the authors argue for a "complementary definition of the duality group" via the orbifold fundamental group of the scalar manifold. However, they also point out that in general only a quotient Γ′ of the full duality group Γ acts on the moduli space. In this case the orbifold fundamental group of the scalar manifold does not necessarily agree with the full duality group. They also provide explicit string theory examples for this situation later (see section 3.4). I therefore believe that eq. (8) does not provide a sensible definition of the duality group Γ unless Γ can be uniquely recovered from its quotient Γ′.
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The motivation following the definition in eq. (4) could be made a bit clearer. Are continuous duality groups also conceivable and is focusing on discrete duality groups just a choice of the authors, or are all sensible duality groups necessarily discrete?
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There is a small typo below eq. (28): Kn(n) should probably read En(n).
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Strengths
The paper presents an original idea: that the compactifiability of moduli spaces—a geometric condition—arises naturally from finiteness constraints on QG amplitudes. This connects a seemingly technical requirement (volume growth control) to a rich structure of string theory dualities, providing a new organizing principle within the Swampland program. The authors successfully combine mathematical ideas (orbifold fundamental groups, semisimple representations) with physically motivated conjectures, drawing on tools from algebraic geometry, and representation theory. Despite the abstract nature of the topic, the manuscript is clearly written and includes well-structured examples, particularly the detailed discussion of SL(2,Z) duality in Type IIB string theory. Concepts like the “marked moduli space,” “no-minimum-length conjecture,” and semisimplicity are explained with care and context.
Weaknesses
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Requested changes
• Clarify footnote 1 regarding the role of GL+(2,Z) vs. SL(2,Z).
• Expand the explanation in Section 4.2 on the relationship between harmonic forms and infinite towers in 1D SUSY QM.
Recommendation
Ask for minor revision